138 ON THE EFFECT OF TUB INTERNAL FRICTION OF FLUIDS
nature and the density of the gas are changed. In the case of a long cylinder, the expression for k is complicated, and involves infinite series. Nevertheless the numerical table on p. 52 shows that if we except the very early part of the table, which corresponds to very slender rods, or (according to Maxwell's law) to very low pressures, the value of k nearly fits an expression of the form (a), A being in this case 13 instead of £, which was its value for a sphere, and B depending, as it did before, on the diameter and time of vibration. And if the pendulum be made up of a sphere and a cylindrical rod, as was the case with many of Baily's, we shall still not go far wrong if we take k to be expressed in the same form (a), in which the constants A and £ admit of being obtained by calculation. Now by the definition of k the effect of the air on the time of vibration of a given pendulum varies as (! + &)/>. Baily's n, or 1 + &, was got by dividing the observed difference in the time of vibration at the high and low pressure, corrected for everything but the effect of the air, by the calculated difference for buoyancy alone.
Hence n=l + A/jp/A/o, where A denotes the difference at the high and low pressures. If we assume k to be given nearly enough by the formula (a), we have
If we put BC for n — 1 — A, we have according to the assumed law vV == @> an(l therefore if \ve denote the higher and lower densities by p13 p0, we have at the atmospheric density
which gives what p is supposed to be. Jjut we ought to have taken
so that if we denote the apparent coefficient, by '/JL, reserving p, for the true coefficient, we have
/ft = _^£__ = A - P» = i , /5 V > VP.A VP V/>, (^ - Vp0) V pl '
Now in the same experiment the swings at high and low pressure were taken at temperatures that did not much differ;